Rotating machinery support excitations can occur if a machine is installed on a base prone to ground motions or on-board moving systems such as ships and aircraft. This paper presents a formulation for the dynamic analysis of rigid rotors subject to base excitations plus mass imbalance. The formulation allows for six motions at the machine base and takes into account the linear/nonlinear spring characteristics of the supporting bearings. Equations of motion are derived using Lagrange’s equations. For rotor—linear bearing systems subject to mass imbalance plus harmonic excitations along or around lateral directions, analytical solutions for equations of motion are derived and analytical results in the time domain are compared with their counterparts obtained by numerical integration using the Runge—Kutta method and typical agreement is obtained. The system natural frequencies as affected by rotor speed are obtained using the QR algorithm using the DAMRO-1 program and compared with those obtained by MATLAB and excellent agreement is obtained. The frequency response (maximum amplitude of vibrations against the base excitation frequency) is characterized by peaks at natural frequencies of the rotating gyroscopic system. This necessitates extreme precaution when we design such rotating systems that are prone to base motions and mass imbalance. For systems with bearing cubic nonlinearity, results are obtained by numerical integration and discussed with regards to the time domain, fast Fourier transform (FFT) and Poincaré map. Periodic and quasi-periodic disk/bearings motions are observed. For systems with support cubic nonlinearity and subject to mass imbalance and base excitation, the FFT of disk horizontal and vertical vibrations is marked with sum and difference tones, ± nfb± fs( n + m is always odd) where fs is the rotating unbalance frequency and fb is base excitation frequency.
A lagrangian formulation is presented for the total dynamic stiffness and damping matrices of a rigid rotor carrying noncentral rigid disk and supported on angular contact ball bearings (ACBBs). The bearing dynamic stiffness/damping marix is derived in terms of the bearing motions (displacements/rotations) and then the principal of virtual work is used to transfer it from the bearing location to the rotor mass center to obtain the total dynamic stiffness/damping matrix. The bearing analyses take into account the bearing nonlinearities, cage rotation and bearing axial preload. The coefficients of these time-dependent matrices are presented analytically. The equations of motion of a rigid rotor-ACBBs assembly are derived using Lagrange's equation. The proposed analyses on deriving the bearing stiffness matrix are verified against existing bearing analyses of SKF researchers that, in turn, were verified using both SKF softwares/experiments and we obtained typical agreements. The presented total stiffness matrix is applied to a typical grinding machine spindle studied experimentally by other researchers and excellent agreements are obtained between our analytical eigenvalues and the experimental ones. The effect of using the total full stiffness matrix versus using the total diagonal stiffness matrix on the natural frequencies and dynamic response of the rigid rotor-bearings system is studied. It is found that using the diagonal matrix affects natural frequencies values (except the axial frequency) and response amplitudes and pattern and causes important vibration tones to be missig from the response spectrum. Therefore it is recommended to use the full total stiffness matrix and not the diagonal matrix in the design/vibration analysis of these rotating machines. For a machine spindle-ACBBs assembly under mass unbalnce and a horizontal force at the spindle cutting nose when the bearing time-varying stiffness matrix (bearing cage rotation is considered) is used, the peak-to-valley variation in time domain of the stiffness matrix elements becomes significant compared to its counterpart when the bearing standard stiffness matrix (bearing cage rotation is neglected) is used. The vibration spectrum of the time-varying matrix case is marked by tones at bearing outer ring ball passing frequency, rotating unbalnce frequency and combination compared to spectrum of the standard stiffness matrix case which is marked by only the rotating unbalnce frequency. Therfore, it is highly recomended to model bearing stiffness matrix to be a time-dependent.
The radial clearance in rolling bearing systems, required to compensate for dimensional changes associated with thermal expansion of the various parts during operation, may cause dimensional attrition and comprise bearing life, if unloaded operation occurs and balls skid [D. Childs and D. Moyer, ASME J. Eng. Gas Turb. Power 107, 152-159 (1985)]. Also, it can cause jumps in the response to unbalance excitation. These undesirable effects may be eliminated by introducing two or more loops into one of the bearing races so that at least two points of the ring circumference provide a positive zero clearance [D. Childs, Handbook of Rotordynamics, edited by F. Ehrich (McGraw-Hill, NY, 1992)]. The deviation of the outer ring with two loops, known as ovality, is one of the bearing distributed defects. Although this class of imperfections has received much work, none of the available studies has simulated the effect of the outer ring ovality on the dynamic behavior of rotating machinery under rotating unbalance with consideration of ball bearing nonlinearities, shaft elasticity, and speed of rotation. To fill this gap, the equations of motion of a rotor-ball bearing system are formulated using finite-elements (FE) discretization and Lagrange's equations. The analyses are specialized to a rigid-rotor system, by retaining the rigid body modes only in the FE solution. Samples of the results are presented in both time domain and frequency domain for a system with and without outer ring ovality. It is found that with ideal bearings (no ovality), the vibration spectrum is qualitatively and quantitatively the same in both the horizontal and vertical directions. When the ring ovality is introduced, however, the spectrum in both orthogonal planes is no longer similar. And magnitude of the bearing load has increased in the form of repeated random impacts, between balls and rings, in the horizontal direction (direction of maximum clearance) compared to a continuous contact along the vertical direction (direction of positive zero clearance). This underlines the importance of the vibration measuring probe's direction, with respect to the outer ring axes, to capture impact-induced vibrations. Moreover, when the harmonic excitation is increased for a system with ideal bearings, the spectral peaks above forcing frequency have shifted to a higher-frequency region, indicating some sort of a hard spring mechanism inherent in the system. Another observation, is that for the same external excitation, vibration amplitude at forcing frequency in the bearing force spectrum is the same for systems with or without outer ring ovality.
An analytical model is presented for studying the effect of tooth backlash and ball-bearing deadband clearance (clearance between bearing outer ring and its housing) on vibration spectrum of spur gear boxes. Included in the model are coupled torsional-lateral vibrations of the shafts, time-varying tooth backlash (backlash is a function of operating center distance), mesh stiffness (stiffness is calculated based on the bending, shear, axial, and Hertzian contact strain energies of the tooth that is treated as a cantilever of involute shape), and Hertzian damping at mesh, friction between meshing teeth, and interaction between gear box casing and internals. The bearing–housing reaction forces are calculated with consideration of bearing clearance and system vibrations. The above parameters are accounted for with allowance for number of teeth pairs sharing the load to alternate between one and two. The analysis is applied to a single stage gear box and equations of motion are deduced using Newton’s laws of motion and numerically integrated using Runge–Kutta method. The gear box casing velocities in time domain are transformed into frequency domain (vibration spectrum) using Fast Fourier Transform (FFT). This was done for different values of tooth backlash and bearing clearance and samples of the results are shown. This model provides an analysis procedure for predicting effect of tooth wear (wear affects backlash) and bearing deadband clearance on gear box vibration spectrum.
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